Finite Group DatabasebetaFinite Group Databasebeta| order | 504 |
| isoclinism | isoclinic class of Dic42⋊C3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | Dic42⋊C3 |
| Hall | C3×ΓL(1,𝔽8) ⋊ Q8 |
| Sylow | C7 ⋊ C3×Dic6 |
54 conjugacy classes of subgroups.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 168 |
A minimal faithful permutation representation — 2 generators on 18 points:
(1,2,3)(4,5,10)(7,8,9)(11,13,14,18)(12,15,16,17);(2,3)(5,6)(7,8)(9,10)(11,12,14,16)(13,17,18,15)Polycyclic presentation:
2^2,3,2,3,7|a^2=d,b^2=d,b^a=bd,e^a=e^2,f^a=f^6,f^c=f^239 conjugacy classes — view the full character table.
| maximal subgroup of | #66565⋊C3, #66567⋊C3, #66570⋊C3, … |
| maximal normal subgroup of | #66565⋊C3, #66567⋊C3, #66570⋊C3, … |
| maximal quotient of | #66629⋊C3, C4⋊(C3⋊(ΓL(1,𝔽8)⋊C4)), C2×Dic42⋊C3, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(504, 85);